3.222 \(\int \frac {x (c+d x^2)^3}{a+b x^2} \, dx\)

Optimal. Leaf size=87 \[ \frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {d x^2 (b c-a d)^2}{2 b^3}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b} \]

[Out]

1/2*d*(-a*d+b*c)^2*x^2/b^3+1/4*(-a*d+b*c)*(d*x^2+c)^2/b^2+1/6*(d*x^2+c)^3/b+1/2*(-a*d+b*c)^3*ln(b*x^2+a)/b^4

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Rubi [A]  time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \[ \frac {d x^2 (b c-a d)^2}{2 b^3}+\frac {\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac {\left (c+d x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(d*(b*c - a*d)^2*x^2)/(2*b^3) + ((b*c - a*d)*(c + d*x^2)^2)/(4*b^2) + (c + d*x^2)^3/(6*b) + ((b*c - a*d)^3*Log
[a + b*x^2])/(2*b^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rubi steps

\begin {align*} \int \frac {x \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+d x)^3}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {d (b c-a d)^2 x^2}{2 b^3}+\frac {(b c-a d) \left (c+d x^2\right )^2}{4 b^2}+\frac {\left (c+d x^2\right )^3}{6 b}+\frac {(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 82, normalized size = 0.94 \[ \frac {b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(b*d*x^2*(6*a^2*d^2 - 3*a*b*d*(6*c + d*x^2) + b^2*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4)) + 6*(b*c - a*d)^3*Log[a +
b*x^2])/(12*b^4)

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fricas [A]  time = 0.46, size = 120, normalized size = 1.38 \[ \frac {2 \, b^{3} d^{3} x^{6} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

1/12*(2*b^3*d^3*x^6 + 3*(3*b^3*c*d^2 - a*b^2*d^3)*x^4 + 6*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 6*(b
^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(b*x^2 + a))/b^4

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giac [A]  time = 0.29, size = 124, normalized size = 1.43 \[ \frac {2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

1/12*(2*b^2*d^3*x^6 + 9*b^2*c*d^2*x^4 - 3*a*b*d^3*x^4 + 18*b^2*c^2*d*x^2 - 18*a*b*c*d^2*x^2 + 6*a^2*d^3*x^2)/b
^3 + 1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(abs(b*x^2 + a))/b^4

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maple [A]  time = 0.00, size = 149, normalized size = 1.71 \[ \frac {d^{3} x^{6}}{6 b}-\frac {a \,d^{3} x^{4}}{4 b^{2}}+\frac {3 c \,d^{2} x^{4}}{4 b}+\frac {a^{2} d^{3} x^{2}}{2 b^{3}}-\frac {3 a c \,d^{2} x^{2}}{2 b^{2}}+\frac {3 c^{2} d \,x^{2}}{2 b}-\frac {a^{3} d^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}-\frac {3 a \,c^{2} d \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {c^{3} \ln \left (b \,x^{2}+a \right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

1/6*d^3/b*x^6-1/4*d^3/b^2*x^4*a+3/4*d^2/b*x^4*c+1/2*d^3/b^3*x^2*a^2-3/2*d^2/b^2*x^2*a*c+3/2*d/b*x^2*c^2-1/2/b^
4*ln(b*x^2+a)*a^3*d^3+3/2/b^3*ln(b*x^2+a)*a^2*c*d^2-3/2/b^2*ln(b*x^2+a)*a*c^2*d+1/2/b*ln(b*x^2+a)*c^3

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maxima [A]  time = 1.06, size = 119, normalized size = 1.37 \[ \frac {2 \, b^{2} d^{3} x^{6} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

1/12*(2*b^2*d^3*x^6 + 3*(3*b^2*c*d^2 - a*b*d^3)*x^4 + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x^2)/b^3 + 1/2*(
b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(b*x^2 + a)/b^4

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mupad [B]  time = 0.06, size = 123, normalized size = 1.41 \[ x^2\,\left (\frac {3\,c^2\,d}{2\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b^4}+\frac {d^3\,x^6}{6\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(c + d*x^2)^3)/(a + b*x^2),x)

[Out]

x^2*((3*c^2*d)/(2*b) + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/(2*b)) - x^4*((a*d^3)/(4*b^2) - (3*c*d^2)/(4*b)) - (log
(a + b*x^2)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(2*b^4) + (d^3*x^6)/(6*b)

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sympy [A]  time = 0.49, size = 94, normalized size = 1.08 \[ x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{2} \left (\frac {a^{2} d^{3}}{2 b^{3}} - \frac {3 a c d^{2}}{2 b^{2}} + \frac {3 c^{2} d}{2 b}\right ) + \frac {d^{3} x^{6}}{6 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

x**4*(-a*d**3/(4*b**2) + 3*c*d**2/(4*b)) + x**2*(a**2*d**3/(2*b**3) - 3*a*c*d**2/(2*b**2) + 3*c**2*d/(2*b)) +
d**3*x**6/(6*b) - (a*d - b*c)**3*log(a + b*x**2)/(2*b**4)

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